A144698 → (θ+1)²

nonnegative coefficients

Proved for all n all roots real, in (−∞, 0]coefficients of one sign Hadamard kernel interlacing (weak)

Recurrence

\(\left(3 + n\right)\,P_{n}(x) = \left(-32\,x + 204\,x^{2} - 24\,n x + 224\,n x^{2} - 4\,n^{2} x + 79\,n^{2} x^{2} + 9\,n^{3} x^{2}\right)\,P_{n-2}(x) + \left(-8\,x + 65\,x^{2} - 57\,x^{3} - 2\,n x + 32\,n x^{2} + 11\,n x^{3} + 4\,n^{2} x^{2} + 22\,n^{2} x^{3} + 4\,n^{3} x^{3}\right)\,P_{n-2}'(x) + \left(-4\,x^{2} + 49\,x^{3} - 45\,x^{4} - n x^{2} + 28\,n x^{3} - 27\,n x^{4} + 4\,n^{2} x^{3} - 4\,n^{2} x^{4}\right)\,P_{n-2}''(x) + \left(3 + 36\,x + n + 24\,n x + 4\,n^{2} x\right)\,P_{n-1}(x) + \left(7\,x - 48\,x^{2} + 2\,n x - 28\,n x^{2} - 4\,n^{2} x^{2}\right)\,P_{n-1}'(x)\)

\(P_{0}(x) = 1\)

\(P_{1}(x) = 1 + 16\,x\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
0	1								1
1	1	16							17
2	1	52	144						197
3	1	128	1017	1024					2170
4	1	284	4779	13136	6400				24600
5	1	600	18810	103520	134625	36864			294420
6	1	1236	67230	648160	1643625	1190484	200704		3751440
7	1	2512	226719	3548800	15307375	21343824	9506833	1048576	50984640

All 15 rows

Technical details

Node id: n-f111e963d66b0e4a · revision 0 · JSON
Definition source: derived:theta_plus_one_squared
Fingerprint: f6-2f639303bdfeec17

Observed on the stored rows ?

Degrees: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14 (affine: slope 1, intercept 0)
Palindromic rows: no
γ-positive (gamma_positive): no
Nonnegative coefficients (nonn): yes
Only rational roots (rr_by_factors: every nonzero row is a product of linear factors over ℚ): no

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x

%e Example

Triangle begins:
  0: 1;
  1: 1,    16;
  2: 1,    52,     144;
  3: 1,   128,    1017,     1024;
  4: 1,   284,    4779,    13136,      6400;
  5: 1,   600,   18810,   103520,    134625,      36864;
  6: 1,  1236,   67230,   648160,   1643625,    1190484,     200704;
  7: 1,  2512,  226719,  3548800,  15307375,   21343824,    9506833,    1048576;
  8: 1,  5068,  736677, 17822704, 120896875,  282361284,  240855727,   70474048,   5308416;
  9: 1, 10184, 2335464, 84387296, 855116150, 3086899704, 4376265656, 2451314816, 493744329, 26214400;

%F Formula

The row polynomials satisfy (n+3)*P(n,x) = -((4*n^2+24*n+32)*x-(9*n^3+79*n^2+224*n+204)*x^2)*P(n-2,x) - ((2*n+8)*x-(4*n^2+32*n+65)*x^2-(4*n^3+22*n^2+11*n-57)*x^3)*d/dx P(n-2,x) - ((n+4)*x^2-(4*n^2+28*n+49)*x^3+(4*n^2+27*n+45)*x^4)*d^2/dx^2 P(n-2,x) + (n+3+(4*n^2+24*n+36)*x)*P(n-1,x) + ((2*n+7)*x-(4*n^2+28*n+48)*x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1 + 16*x.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 1 + 16*x;
P[n_, x_] := P[n, x] = Expand[Together[(-((4*n^2 + 24*n + 32)*x - (9*n^3 + 79*n^2 + 224*n + 204)*x^2)*P[n - 2, x] - ((2*n + 8)*x - (4*n^2 + 32*n + 65)*x^2 - (4*n^3 + 22*n^2 + 11*n - 57)*x^3)*D[P[n - 2, x], x] - ((n + 4)*x^2 - (4*n^2 + 28*n + 49)*x^3 + (4*n^2 + 27*n + 45)*x^4)*D[P[n - 2, x], {x, 2}] + (n + 3 + (4*n^2 + 24*n + 36)*x)*P[n - 1, x] + ((2*n + 7)*x - (4*n^2 + 28*n + 48)*x^2)*D[P[n - 1, x], x])/(n + 3)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 1 + 16*x; for(n = 2, N, my(i = n + 1); v[i] = (-((4*n^2 + 24*n + 32)*x - (9*n^3 + 79*n^2 + 224*n + 204)*x^2)*v[i - 2] - ((2*n + 8)*x - (4*n^2 + 32*n + 65)*x^2 - (4*n^3 + 22*n^2 + 11*n - 57)*x^3)*deriv(v[i - 2], x) - ((n + 4)*x^2 - (4*n^2 + 28*n + 49)*x^3 + (4*n^2 + 27*n + 45)*x^4)*deriv(deriv(v[i - 2], x), x) + (n + 3 + (4*n^2 + 24*n + 36)*x)*v[i - 1] + ((2*n + 7)*x - (4*n^2 + 28*n + 48)*x^2)*deriv(v[i - 1], x))/(n + 3)); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i - 1, v[i])))};
T(9)

polytool c48629374597

Properties

Real-rooted? Every row P_n, n ≥ 0, is zero or real-rooted with all roots in (−∞, 0]
by Laguerre’s theorem on multiplier sequences
Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
derived from Laguerre’s theorem on multiplier sequences
Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
derived from Laguerre’s theorem on multiplier sequences
Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, unordered) for every m ≥ 1
by Laguerre’s theorem on multiplier sequences

Proof chains ? JSON

Laguerre’s theorem on multiplier sequences [SP14; Bra15, Thm 7.1 of arXiv:1410.6601] polytool ? step ›
- from Triangle of 4-Eulerian numbers.?
  - Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ? step › direct check

References

[Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
[LW07] Lily L. Liu and Yi Wang. A unified approach to polynomial sequences with only real zeros. Advances in Applied Mathematics, 38(4):542–560, May 2007. doi:10.1016/j.aam.2006.02.003
[SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.